Approximating the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e331" altimg="si11.svg"><mml:mi>p</mml:mi></mml:math>th root by composite rational functions

A landmark result from rational approximation theory states that x1∕p on [0,1] can be approximated by a type-(n,n) function with root-exponential accuracy. Motivated the recursive optimality property of Zolotarev functions (for square root and sign functions), we investigate approximating composite form rk(x,rk−1(x,rk−2(⋯(x,r1(x,1))))). While this class ceases to contain minimax (best) approximant for p≥3, show it achieves approximately pth-root exponential convergence respect degree. Moreover, crucially, is doubly number degrees freedom, suggesting are able approximate related (such as |x| sector function) exceptional efficiency.